R E S E A R C H
Open Access
Pinning control design for feedback
stabilization of constrained Boolean control
networks
Qiqi Yang, Haitao Li
*and Yansheng Liu
*Correspondence:
[email protected] School of Mathematical Science, Shandong Normal University, Jinan, 250014, P.R. China
Abstract
We study the pinning control design for feedback stabilization of constrained Boolean control networks (BCNs) via the semitensor product of matrices. Firstly, a constrained algebraic representation is obtained for constrained BCNs with pinning control, and a necessary and sufficient condition is established for the reachability of constrained BCNs with pinning control. Secondly, a general procedure is proposed for the pinning control design of state feedback stabilization of constrained BCNs. Thirdly, a necessary and sufficient condition is presented for the output feedback stabilization of
constrained BCNs with pinning control. Finally, two illustrative examples are worked out to demonstrate the effectiveness of the obtained new results.
MSC: 93C55; 93B52
Keywords: Boolean control network; pinning control; feedback stabilization; constrained control; semitensor product of matrices
1 Introduction
In recent years, with the development of systems biology and medical science, there have been a lot of computational models to simulate gene regulatory networks (GRNs) [, ]. In , Kauffman [] firstly proposed the concept of Boolean networks to model GRNs. From then on, the study of Boolean networks has attracted attention of many scholars [–]. In a Boolean network, genes are simulated as and for studying the activity of genes. Since the dynamics of Boolean networks does not contain parameters, we can use Boolean networks to model large-scale GRNs.
Recently, a semitensor product method [] has been proposed for the study of Boolean networks. Using this novel method, we can convert a Boolean (control) network into a linear (bilinear) form. Then, we can conveniently investigate Boolean networks with the help of classical control theory. This framework is called the algebraic state space rep-resentation (ASSR) of logical systems. Up to now, many scholars have applied the ASSR framework to the analysis and control of Boolean networks and obtained lots of inter-esting results [–]. Moreover, the semitensor product method has also been applied to networked evolutionary games [, ] and feedback shift registers [].
As one of the most important issues in the study of GRNs, the stabilization of Boolean control networks (BCNs) has been found to be widely applied in the design of therapeutic
interventions and the explanation of some living phenomena []. The state feedback sta-bilization of BCNs was studied in [, ], and a novel design procedure was established. Later, the design of output feedback stabilizers of BCNs was investigated [–], and some necessary and sufficient conditions were presented.
Note that the pinning control problem has been introduced to the study of Boolean net-works in some recent net-works [, ]. With the pinning control, we can realize the control objective by controlling a small number of nodes. Using the ASSR framework, the pin-ning control design for the stabilization of Boolean networks was studied in [], and a novel design procedure was established. Moreover, in GRNs, some states and inputs are undesirable because they may lead to a dangerous situation such as the deterioration of a disease, the metastasis of a cancer [, ], and so on. Hence, we need put some constraints to the undesirable states and inputs in BCNs. It should be noticed that the pinning control design for feedback stabilization of constrained BCNs is still a challenging problem, and there are fewer results on this problem. In addition, the pinning stabilization control de-sign procedure proposed in [] cannot be directly applied to the pinning control dede-sign for feedback stabilization of constrained BCNs. This motivates us to study the pinning control design for feedback stabilization of constrained BCNs.
In this paper, using the ASSR framework, we investigate the pinning control design for feedback stabilization of constrained BCNs and present a number of new results. The main contributions of this paper are as follows. On one hand, we convert the dynamics of constrained BCNs with pinning control into a constrained algebraic form, which facilitates the reachability analysis and control design of constrained BCNs with pinning control. On the other hand, we present some necessary and sufficient conditions for the pinning control design of feedback stabilizers of constrained BCNs, which can be easily verified with the help of MATLAB.
The rest of this paper is organized as follows. In Section , we give some preliminaries on the semitensor product of matrices. In Section , we investigate the pinning control design for feedback stabilization of constrained BCNs and present the main results of this paper. Two illustrative examples are worked out to verify the obtained new results in Section , which is followed by a brief conclusion in Section .
Notation R,N, andZ+denote the sets of real numbers, natural numbers, and positive
integers, respectively;D:={, },Dn:=D× · · · ×D
n
,n:={δnk|≤k≤n}, whereδnk
de-notes thekth column ofIn, and for compactness,:=. Ann×tmatrixMis called a
logical matrix ifM= [δi
n δin · · · δint]. We expressMbriefly asM=δn[ii · · · it]. Denote
the set ofn×tlogical matrices byLn×t. Ann×tmatrixM= (Mi,j) is called a Boolean
matrix ifMi,j∈D,i= , . . . ,n,j= , . . . ,t. Denote the set ofn×tBoolean matrices byBn×t.
Given a real matrixA∈Rm×n,Col
i(A) denotes theith column ofA, andRowi(A) denotes
theith row ofA; ‘∗’ denotes the Khatri-Rao product of matrices.
2 Preliminaries
First of all, we give some fundamental preliminaries on the semitensor product of matrices. For details, we refer to [].
Definition ([]) The semitensor product of two matricesA∈Rm×nandB∈Rp×qis
AB= (A⊗Iα
whereα=lcm(n,p) represents the least common multiple ofnandp, and⊗is the Kro-necker product.
Obviously, whenn=p, the semitensor product ofAandBbecomes the conventional matrix product. Thus, we can simply call it ‘product’ and omit the symbol ‘’ if no confu-sion arises.
Proposition ([]) Let X∈Rt×be a column vector,and A∈Rm×n.Then
XA= (It⊗A)X. ()
Proposition ([]) Let X∈Rm×and Y∈Rn×be two column vectors.Then
YX=W[m,n]XY, ()
where W[m,n]∈Lmn×mnis the so-called swap matrix defined as
W[m,n]=δmn
m+ · · · (n– )m+ m+ · · · (n– )m+
· · ·
m m+m · · · (n– )m+m.
Proposition ([]) Define the matrix,Mr,k,called the k-valued power-reducing matrix,
as
Mr,k=
⎡ ⎢ ⎢ ⎢ ⎢ ⎣
δ
k k · · · k
k δk · · · k
.. .
k k · · · δkk
⎤ ⎥ ⎥ ⎥ ⎥
⎦, ()
wherek∈Rkis the zero vector.
Denoting ∼δand ∼δ, we have∼D, where ‘∼’ denotes two different forms of the same object. In most places of this paper, we useδ
andδto express logical variables
and call them the vector form of logical variables. The following lemma is fundamental for the matrix expression of logical functions.
Lemma ([]) Let f(x,x, . . . ,xs) :Ds→D be a logical function. Then,there exists a
unique matrix Mf∈L×s,called the structural matrix of f,such that
f(x,x, . . . ,xs) =Mfx· · ·xs:=Mfsi=xi, xi∈. ()
For example, the structural matrices for negation (¬), conjunction (∧), disjunction (∨), conditional (→), biconditional (↔), and exclusive Or (¯∨) areMn=δ[ ],Mc=δ[ ], Md=δ[ ],Mi=δ[ ],Me=δ[ ] andMp=δ[ ], respectively.
3 Main results
In this section, we give the main results of this paper. Firstly, we convert the dynamics of a constrained Boolean network with pinning control into its equivalent constrained algebraic form, based on which we obtain a necessary and sufficient condition for the reachability of constrained BCNs with pinning control. Secondly, we present the pinning control design procedure for the feedback stabilization of constrained BCNs.
3.1 Constrained algebraic form
A Boolean network withnnetwork nodes andrpinning controls can be described as
and
y(t) =Hx(t)Hx(t)· · ·Hpx(t)
=H∗H∗ · · · ∗Hpx(t)
:=Hx(t), ()
where
L=F
r
i=
(In+i–⊗Fi)W[n,n+i–]Mr,n∈Lr×n+r,
Mr,n=diagδ
n,δn, . . . ,δ n
n
∈Ln×n,
L=Fr+∗Fr+∗ · · · ∗Fn∈Ln–r×n,
and
H=H∗H∗ · · · ∗Hp∈Lp×n.
Summarizing, we obtain the following algebraic form of system ():
⎧ ⎨ ⎩
x(t+ ) =Qu(t)x(t),
y(t) =Hx(t), ()
whereQ=L(In+r⊗L)W[n,n+r]Mr,nW[r,n]∈Ln×n+r.
Now, we consider system () with state and input constraints. For anyt∈N, we assume
thatx(t)∈Sx⊆ n andu(t)∈Su⊆ r. Let|Sx|=n≤nand|Su|=r≤r, where|Sx| denotes the cardinality of the setSx. Then,SxandSucan be expressed as
Sx=
δik
n:k= , . . . ,n; ≤i<· · ·<in≤
n, ()
Su=
δjk
r:k= , . . . ,r; ≤j<· · ·<jr≤
r. ()
Denote the trajectory of system () with a pinning control sequence {(u(t),u(t), . . . , ur(t)) :t∈N} ⊆Suand an initial statex∈Sxbyx(t;x, (u(t),u(t), . . . ,ur(t))).
In the following, we convert system () with state and input constraints into an equiv-alent constrained algebraic form.
Define the following set of matrices:
Ji(p,q):= [q×q · · · q×q Iq ith
q×q · · · q×q
p
], ()
whereJi(p,q)∈Rq×pq,i= , , . . . ,p,
q×qdenotes theq×qzero matrix, andIq∈Lq×qis the
Proposition ([]) .Given a matrix A∈Rpq×r,split A as
r of the constrained system into the following form:
Remark SinceQis a logical matrix, we can easily conclude that each column ofQhas at most one element ‘’.
Based on the above transformation, we convert system () into the following form: ⎧
⎨ ⎩
x(t+ ) =Qu(t)x(t),
y(t) =Hx(t). ()
Proposition The state trajectories of system()with Sxand Suare equivalent to that
of system()withSxandSu.
Proof On one hand,∀t∈N,∀u(t) =δjsr∈Su, and∀x(t) =δiαn∈Sx, ifx(t+ ) =Qu(t)x(t)∈
Sx, say,x(t+ ) =δ iβ
n,β∈ {, . . . ,n}, a simple calculation shows that
x(t+ ) =Qu(t)x(t) =δnβ
∈Sx\
δn
,
wherex(t) =δα
n∈Sxandu(t) =δ
s
r∈Su. Ifx(t+ ) =Qu(t)x(t) =δ
i
n∈/ Sx, thenx(t+ ) =
Qu(t)x(t) =δ
n. Hence, in both cases,x(t+ ) =xx(t+ ). On the other hand,∀t∈N,∀u(t) =δs
r∈Su\ {δ
r}, and∀x(t) =δ
α
n∈Sx\ {δ
n}, ifx(t+ ) =
Qu(t)x(t)∈Sx\ {δn}, say,x(t+ ) =δ
β
n,β∈ {, . . . ,n}, we can obtain that
x(t+ ) =Qu(t)x(t) =δiβn∈Sx.
Ifx(t+ ) =Qu(t)x(t) =δn, then
x(t+ ) =Qu(t)x(t) /∈Sx.
Hence, we havex(t+ ) =xx(t+ ).
Therefore, the state trajectories of system () withSxandSuare equivalent to that of
system () withSxandSu.
Remark We call () the constrained algebraic form of the original system. Based on Proposition , we can convert the feedback stabilization of the original system to that of system ().
3.2 Reachability analysis
In this subsection, we study the reachability of system (), which is crucial to pinning control design for the feedback stabilization.
We give the definition of the reachability for system () as follows.
Definition For system (), given two statesx,xd∈Sx\ {δn} and a integer k> , xd is said to be reachable fromx at time k if there is a pinning control sequence
(u(t),u(t), . . . ,ur(t)) with u(t) = ir=ui(t)∈Su \ {δr}, t ∈ {, , . . . ,k – }, such that x(k;x, (u(t),u(t), . . . ,ur(t))) =xd.
For system (), consider two given statesx=δαn∈Sx\ {δ
n},xd=δ
β
n∈Sx\ {δn}and a given integerk> . LetP(k;x,xd) denote the number of different paths such thatxdis
Lemma Consider system()with two given statesx=δnα∈Sx\ {δ
Proof We prove this lemma by induction. Firstly, lettingk= , assume thatu=δρ
r} be different control sequences such thatxcannot reach xd in one step. Hence, it is easy to see that (Qρl)β,α= , ∀l∈ {, . . . ,s}, and (Qρl)β,α= ,
By induction, () holds for any integerk> . This completes the proof.
Based on Lemma , we give the following result on the reachability of system ().
Remark From the proof of Theorem we easily see that
P(k;x,xd) =
δβnT(Q)kδnα=Qkβ,α.
All of them denote the number of different paths such thatxdis reachable fromxat
timek.
3.3 Feedback stabilization pinning control design
In this part, we study the pinning control design for the feedback stabilization of con-strained BCNs. By Proposition we consider the feedback stabilization of system () based on the reachability analysis.
Firstly, we give the definition of stabilization for system () withSxandSu.
Definition System () with Sx and Su is said to be stabilizable to a given
equi-librium xe ∈Sx if there exists a pinning control sequence {(u(t),u(t), . . . ,ur(t)) :t ∈
N} ⊂ Su under which the trajectory initialized at any x ∈ Sx converges to xe and
x(t;x, (u(t),u(t), . . . ,ur(t)))∈Sx,∀t∈N.
In this paper, we study the following two kinds of feedback pinning controls:
. State feedback pinning control:
ui(t) =Kix(t), ()
whereKi∈L×n,i= , . . . ,r.
. Output feedback pinning control:
ui(t) =Giy(t), ()
whereGi∈L×p,i= , . . . ,r.
In the following, we consider the pinning control design for the state feedback stabiliza-tion of system () withSxandSubased on the constrained algebraic form.
For system (), letxe=xxe=δnα∈Sx\ {δ
n}. For any integerk> , define
k(xe) =
δβn
∈Sx\
δn
: there exists a control sequence
u(),u(), . . . ,u(k– )∈Su\
δr
such that
xk;δβn,u(), . . . ,u(k– )=δnαand xl;δnβ,u(), . . . ,u(l– )∈Sx\
δn,∀l∈ {, . . . ,k}. () Proposition System()is stabilized toxe=δnαby a state feedback control if and only
if there exists a positive integerσ≤nsuch that . (Q)α,α> ,
. Rowα(Q
σ
) > .
Proof (Sufficiency) We can see from Condition and Theorem thatxe∈(xe), which
Denote
◦k(xe) =k(xe)\k–(xe), ()
where(xe) :=∅. Then, we obtain◦k(xe)∩
◦
k(xe) =∅,∀k,k∈ {, . . . ,σ},k=k. More-over, by Condition and Theorem we haveσk=◦k(xe) =Sx\ {δn}. Thus, for any integer
isatisfying ≤i≤n, we can find the unique integer ≤ki≤σ such thatδin∈
◦
ki(xe). For system (), setQ=δn[μ,μ, . . . ,μnr]. Whenki= , we can find an integer ≤
ωi≤rsuch thatQδrωiδin=δ
μ(ωi–)n+i
n =xe. When ≤ki≤σ, we can find an integer ≤ωi≤rsuch thatQδrωiδin=δ
μ(ωi–)n+i
n ∈ki–(xe). Let
K=δr[ω,ω, . . . ,ωn]∈Lr×n. ()
Then, for any initial statex=δni∈Sx\ {δ
n}, ifki= , then we obtain x(;x,u) =Qux=QKxx=δ
μ(ωi–)n+i
n =xe; if ≤ki≤σ, we have
x(;x,u) =Qux=QKxx=δ μ(ωi–)n+i
n ∈ki–(xe).
Hence,x(ki;x,u) =xe,∀≤i≤n, andx(t;x,u)∈Sx\ {δn},∀≤t≤ki– . Sincexe∈
(xe), we obtain
x(t;x,u) =xe, ∀t≥σ,∀x∈Sx\
δn,
which implies that system () is stabilized toxe=δαnby the state feedback controlu(t) =
Kx(t).
(Necessity) The proof of this part is based on a straightforward calculation, and thus we
omit it here.
Based on Propositions and , we have the following result.
Theorem System()with Sxand Suis stabilized to xeby a state feedback control if and
only if there exists a positive integerσ≤nsuch that(Q)α,α> andRowα(Q σ
) > .
From the proof of Proposition we get the following procedure for the pinning control design of state feedback stabilization of constrained BCNs.
Remark The procedure contains the following steps:
. Calculatek(xe)andk◦(xe),k= , . . . ,σ.
. For every integer≤i≤n, find the unique integer≤ki≤σsatisfyingδni∈
◦
ki(xe).
. Find an integer≤ωi≤rsuch that ifki= , thenδ
μ(ωi–)n+i
n =xe; ifki≥, then
δμ(ωi–)n+i
. The state feedback pinning control can be designed asui(t) =Kix(t),i= , . . . ,r, with
K∗K∗ · · · ∗Kr=K, whereK=δr[p, . . . ,pn], and ⎧
⎨ ⎩
pt=jωρ ift=iρ,ρ∈ {, . . . ,n}, pt∈ {j, . . . ,jr} otherwise.
()
Finally, we discuss the pinning control design for the output feedback stabilization of constrained BCNs. To this end, we recall the definition of a nilpotent matrix.
Definition A nilpotent matrixNis a square matrix such thatNk= for some positive
integerk. The smallest suchkis called the degree ofN.
For system () with an output feedback controlu(t) =Gy(t),G∈Lr×p, we have
x(t+ ) =Qu(t)x(t) =QGy(t)x(t) =QGHM r,nx(t), ()
whereMr,n=diag{δ
n,δ
n, . . . ,δ
n
n}. Then, we have the following result on the output feed-back stabilization of system ().
Theorem System()is stabilizable toxe=δαnby an output feedback control if and only
if there exist a logical matrixG∈Lr×pand an integer≤τ ≤nsuch that
QGHMr,n= ⎡ ⎢ ⎣
A ζ A
ζ ζ A ζ A
⎤ ⎥
⎦ ()
and
A A A A
()
is a nilpotent matrix of degree τ, where A ∈ B(α–)×(α–), A ∈ B(α–)×(n–α), A ∈ B(n–α)×(α–),A∈B(n–α)×(n–α), ζ andζ are some proper Boolean row vectors, andζ
andζare zero column vectors.
Proof (Sufficiency) Since
A A A A
()
is a nilpotent matrix of degreeτ, we see that
A A A A
t
for any integert≥τ, which, together with a simple calculation, shows that
(QGHMr,n)
t=
⎡ ⎢ ⎣
· · · · · · · · ·
⎤ ⎥
⎦=δn[α,α, . . . ,α] ()
for any integert≥τ.
Hence, we obtain from () that
x(t) = (QGHM r,n)
tx() =x
e ()
for anyx()∈Sx\ {δn}and any integert≥τ.
Therefore, system () is stabilizable toxe=δnαby the output feedback controlu(t) =
Gy(t).
(Necessity) Suppose that system () is stabilizable toxe=δαn by an output feedback control, say,u(t) =Gy(t). Then, we can find the smallest integer ≤τ≤nsuch that ()
holds for anyx()∈Sx\ {δn}and any integert≥τ. Hence, (QGHMr,n)
τ =δ
n[α,α, . . . ,α]. SplitQGHM r,n∈Ln×ninto the following blocks:
QGHMr,n= ⎡ ⎢ ⎣
A ζ A
ζ λ ζ A ζ A
⎤ ⎥ ⎦,
whereλ∈ {, },A∈B(α–)×(α–),A∈B(α–)×(n–α),A∈B(n–α)×(α–),A∈B(n–α)×(n–α),
ζandζare some proper Boolean row vectors, andζandζare some proper Boolean
column vectors.
It is easy to see fromQGHMr,nxe=xethatλ= , andζandζare zero column vectors. In the following, we prove that
A:=
A A A A
is a nilpotent matrix of degreeτ. We can easily see from (QGHMr,n)
τ=δ
n[α,α, . . . ,α] thatA
τ= , which shows thatA
is a nilpotent matrix. If its degree is less thanτ, then there exists a positive integerτ<τ
such thatAτ= , and thus
(QGHMr,n)
τ=δ
n[α,α, . . . ,α],
which is a contradiction to the minimality ofτ. This completes the proof.
Remark Based on Theorem and Algorithm presented in [], we can design an output feedback gain matrix, say, G=δr[w,w, . . . ,wp], under which system () is stabilizable toxe=δαn. Then, the output feedback pinning control can be designed as
Remark It should be pointed out that the pinning control design for the state feedback stabilization of Boolean networks was studied in [], and a novel design procedure was established. Compared with [], our main results have the following advantages: (i) we established some necessary and sufficient conditions for the pinning control design of both state feedback and output feedback stabilization problems, whereas [] only considered the state feedback stabilization problem; (ii) our results are applicable to the pinning con-trol design for the state feedback stabilization of constrained BCNs.
4 Illustrative examples
In the section, we give two illustrative examples to show how to use the obtained results to design pinning control for feedback stabilization of constrained BCNs.
Example Consider the following Boolean model to simulate theλphage, which is a virus growing on a bacterium. The virus can only follow one of two different pathways: lysogeny or lysis, after injecting chromosome into the bacterium cell. The molecular mechanism responsible for the lysogeny/lysis decision is known asλswitch []. For example, two genes, cI and cro, directly affect the decision. When cI is active (inactive) and cro is inac-tive (acinac-tive), the phage is in the lysogenic (lytic) state, and whether lysogenic state will be established or not depends on five phage genes, cI, cro, cII, cIII, N, and the environmental state. More details can be found in []. Its dynamic can be described in the form
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
cII(t+ ) = (¬cI(t))∧(N(t)∨cIII(t))∧u(t),
cIII(t+ ) = (¬cI(t))∧N(t)∧u(t),
N(t+ ) = (¬cI(t))∧(¬cro(t)),
cI(t+ ) = (¬cro(t))∧(cI(t)∨cII(t)),
cro(t+ ) = (¬cI(t))∧(¬cII(t)),
()
whereu(t) is a binary input that represents whether environmental condition is favorable or not.
Lettingx(t) =cII(t)cIII(t)N(t)cI(t)cro(t), we obtain the following algebraic form:
x(t+ ) =Qu(t)x(t), ()
where
Q=δ[
].
favorable, then the genes cII and cIII are not activated, the cro gene remains active, and its product represses the cI gene. Thus, the lytic state is established.
Due to the limitation of the environmental conditions, we constrain the state inSx=
{δ,δ,δ,δ}.
By Theorem and Remark we can design state feedback pinning controls that stabilize system () withSxto the equilibriumδ(the lytic state), and one of them is
u(t) =δ[ ]x(t).
Example Consider the following BCN:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x(t+ ) =x(t)∨u(t),
x(t+ ) = (x(t)→x(t))∨u(t),
x(t+ ) =¬x(t);
y(t) =x(t)∨x(t),
y(t) = (x(t)∧x(t))∨(¬x(t)∧(x(t)↔x(t))).
()
Settingx(t) =i=xi(t),u(t) =i=ui(t), andy(t) =i=yi(t), we obtain the following
al-gebraic form: ⎧ ⎨ ⎩
x(t+ ) =Qu(t)x(t),
y(t) =Hx(t), ()
where
Q=δ[ ]
and
H=δ[ ].
We assume thatSx={δ,δ,δ,δ,δ,δ}, andSu={δ,δ,δ,δ}. We aim to design an
output feedback pinning control such that system () withSxandSuis stabilized toxe=
δ.
Based on Theorem and Algorithm presented in [], we can obtain eight output feedback pinning control gain matrices; one of them isK=δ[ ],K=δ[ ].
5 Conclusion
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
QY and HL carried out the main part of this article, and YL corrected the manuscript and brought forward many suggestions on this article. All authors have read and approved the final manuscript.
Acknowledgements
The authors would like to thank the Associate Editor and the anonymous reviewers for their constructive comments and suggestions, which improved the quality of this paper. The research was supported by the National Natural Science Foundation of China under Grants 61374065 and 61503225, the Major International (Regional) Joint Research Project of the National Natural Science Foundation of China under Grant 61320106011, and the Natural Science Foundation of Shandong Province under Grant ZR2015FQ003.
Received: 6 February 2016 Accepted: 22 June 2016
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